Morita Context Functors And Torsion Theories

Hereditary torsion theories have been developed since the 1960's and have been extensively studied by Golan,Gabriel,Dickson,Stenstrom,etc. In this thesis ,combining hereditary torsion theories with Morita contexts ,we discuss the changes of some properties about hereditary torsion theories by the Morita context functors and the covers and envelopes by a special category .In the first section ,some preliminaries are mentioned .In order to build a good foundation for later parts ,we introduce some basic concepts and some properties which are mainly the results of hereditary torsion theories and Morita contexts .They can be found in [1],[2].In the second section ,we firstly prove the isomorphism of M and M *,and then discuss the relationship beetween RM and sM and the changes of some properties about hereditary torsion theories by the Morita context functors .In the third section , using the properties of cocritical modules by the Morita context functors , we give the concepts of -prime torsion theories .Meanwhile we argue about several properites and the important relationship beetween -prime torsion theories and -prime torsion theories.In the last section ,we discuss the covers and envelopes both of which are of great importance in the field of ring theories .By the Morita context functors we can transform from Rx-cover(envelope) into Sx-f-cover(envelope).Finally they are further characterized by the particular properties of the direct limit.